Population growth The population of a large city grows exponentially with a current population of 1.3 million and a predicted population of 1.45 million 10 years from now.

b. Find the doubling time of the population.

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Recognize that the population growth follows an exponential model of the form \(P(t) = P_0 \cdot e^{kt}\), where \(P_0\) is the initial population, \(k\) is the growth rate, and \(t\) is time in years.

Use the given information to find the growth rate \(k\). You know \(P_0 = 1.3\) million and \(P(10) = 1.45\) million, so set up the equation \(1.45 = 1.3 \cdot e^{10k}\).

Solve for \(k\) by dividing both sides by 1.3 and then taking the natural logarithm: \(\ln\left(\frac{1.45}{1.3}\right) = 10k\), which gives \(k = \frac{1}{10} \ln\left(\frac{1.45}{1.3}\right)\).

Recall that the doubling time \(T\) is the time it takes for the population to double, so \(P(T) = 2P_0\). Substitute into the model: \(2P_0 = P_0 \cdot e^{kT}\), which simplifies to \(2 = e^{kT}\).

Take the natural logarithm of both sides to solve for \(T\): \(\ln(2) = kT\), so \(T = \frac{\ln(2)}{k}\). Use the value of \(k\) found in step 3 to express the doubling time.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Growth Model

Exponential growth describes a process where the quantity increases at a rate proportional to its current value, often modeled by P(t) = P_0 * e^(kt). Here, P_0 is the initial population, k is the growth rate, and t is time. Understanding this model is essential to relate population changes over time.

Doubling Time

Doubling time is the period required for a quantity growing exponentially to double in size. It can be found using the formula T = ln(2)/k, where k is the growth rate. This concept helps determine how quickly the population will reach twice its current size.

Natural Logarithm and Growth Rate Calculation

The natural logarithm (ln) is used to solve for the growth rate k in the exponential model by rearranging P(t) = P_0 * e^(kt). Calculating k involves taking ln of the ratio of populations and dividing by time, which is crucial for finding doubling time and making predictions.

Population growth The population of a large city grows exponentially with a current population of 1.3 million and a predicted population of 1.45 million 10 years from now.b. Find the doubling time of the population.